fWAR (the version you'll find on Fangraphs) and bWAR (the version on baseball-reference) are two different statistics, that measure different things. Despite the fact that they both talk about "wins above replacement" it's probably best not to think of them as rival accounts of a single value.
There are several differences between them, but the biggest one is in how they deal with pitching. Pitching values in fWAR are based on the pitchers' strikeout rate, walk rate, and homerun allowed rate, since these figures together predict future ERA better than does a pitcher's past ERA. In bWAR, pitcher value is based on runs allowed, which does a less-good job at predicting future ERA, but, of course, does a better job at capturing, you know, the runs that the pitcher actually allowed.
One way to think about the difference between the two is that they are answers to two different questions. fWAR is an answer to the question "if this player were to play for a randomly selected team, how many more (or fewer) games should we expect that team to win in the future?", whereas bWAR is an answer to the question "if this player were to have played for a randomly selected team, how many more (or fewer) games should we expect them to have won?". I think that, for hall of fame purposes, bWAR is the one that you want to look at. The hall of fame is a retrospective thing, and it's bWAR that's the retrospective one.
(In fact, there are, IMHO, pretty limited uses for fWAR. If you are trying to predict future performance, you usually won't want to strip out context in the way that fWAR does. You'll probably want to predict performance of the player if he's playing for some specific team, not a randomly chosen one.)
There's nothing subjective about either WAR measure, they're just algorithms that take events that actually happened and spits out a number. Just like ERA and batting average.
Very briefly, here's the general idea. WAR calculations are based on something called "linear weights". The idea behind linear weights is that each event that happens on a baseball field (a batter hitting a single, for example), has a run value. The run value of an event is how many more or fewer runs a team scores, on average, after an event of that kind. Run values are derived historically - you get them by adding up all of the runs that actually scored after someone hit a single (for example), above or below the average number of runs scored in an otherwise similar situation, and dividing by the number of singles that were actually hit. (You can use all of baseball history for this if you want, but it makes more sense to use a limited range of years, since the run value of an event will vary over time.) By adding up the run values of everything that a player did, you get the number of runs he produced. You then divide that number by the number of runs that will, on average, win a game for a team. (About 10 in recent years - note, this isn't runs scored, it's runs scored PLUS runs that you prevented the other team from scoring.) That gives you the number of wins a player produced. You then find the wins produced by "replacement level" players. A replacement level player is the last guy on the roster, the kind of guy who every team has stashed in AAA, the kind of guy who is barely (or almost) a major leaguer. That kind of guy. And then you subtract replacement level wins from the wins that your player produced to get his wins above replacement.
That's the general idea. There are lots of complications involved in, for example, adjusting for the park that a player plays in. But all of these extra adjustments (that I'm skipping over for the sake of this short post) are also derived from historical data. That is, they, just like the linear weights calculations, reflect what actually happened on the baseball field.
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